A cistern can be filled by two pipes in 30 and 40 minutes respectively ,Both the pipes were opned at once ,but after some time the first was shut up and the cistern was filled in 10 minutes more. How long after the pipes had been opned was the first pipe shut up ?

Options

A. (90)/11 minutes
B. (90)/7 minutes
C. (90)/(13) minutes
D. (45)/2 minutes

Answer

To solve the problem, we need to find out how long after the pipes were opened the first pipe was shut off. Let's break it down step by step.

### Step 1: Determine the rates of the pipes
1. **Pipe A** fills the cistern in 30 minutes. Therefore, its rate is:
\[ \text{Rate of Pipe A} = \frac{1}{30} \text{ cisterns per minute} \]
2. **Pipe B** fills the cistern in 40 minutes. Therefore, its rate is:
\[ \text{Rate of Pipe B} = \frac{1}{40} \text{ cisterns per minute} \]

### Step 2: Calculate the combined rate of both pipes
When both pipes are open, their combined rate is:
\[ \text{Combined Rate} = \text{Rate of Pipe A} + \text{Rate of Pipe B} = \frac{1}{30} + \frac{1}{40} \]
To add these fractions, we need a common denominator. The least common multiple of 30 and 40 is 120.
\[ \frac{1}{30} = \frac{4}{120}, \quad \frac{1}{40} = \frac{3}{120} \]
Thus,
\[ \text{Combined Rate} = \frac{4}{120} + \frac{3}{120} = \frac{7}{120} \text{ cisterns per minute} \]

### Step 3: Let \( t \) be the time in minutes that both pipes are open
In \( t \) minutes, the amount of the cistern filled by both pipes is:
\[ \text{Amount filled} = \text{Combined Rate} \times t = \frac{7}{120} t \]

### Step 4: Determine the remaining time to fill the cistern after Pipe A is shut off
After \( t \) minutes, Pipe A is shut off, and only Pipe B continues to fill the cistern. The problem states that the cistern was filled in 10 minutes more after Pipe A was shut off.

During these 10 minutes, Pipe B fills:
\[ \text{Amount filled by Pipe B} = \text{Rate of Pipe B} \times 10 = \frac{1}{40} \times 10 = \frac{10}{40} = \frac{1}{4} \text{ of the cistern} \]

### Step 5: Set up the equation for the total amount filled
The total amount filled by both pipes together and then by Pipe B alone must equal 1 (the full cistern):
\[ \frac{7}{120} t + \frac{1}{4} = 1 \]
To solve for \( t \), first convert \( \frac{1}{4} \) to a fraction with a denominator of 120:
\[ \frac{1}{4} = \frac{30}{120} \]
Now substitute this back into the equation:
\[ \frac{7}{120} t + \frac{30}{120} = 1 \]
Multiply through by 120 to eliminate the denominator:
\[ 7t + 30 = 120 \]
Subtract 30 from both sides:
\[ 7t = 90 \]
Now, divide by 7:
\[ t = \frac{90}{7} \approx 12.86 \text{ minutes} \]

### Step 6: Conclusion
The first pipe was shut off approximately 12.86 minutes after both pipes were opened.